Gain crossover satisfies $|G(j\omega)|=2\sqrt{1+\omega^2}/\omega^2=1$. Squaring gives $\omega^4-4\omega^2-4=0$, so $\omega_c^2=2+2\sqrt2$ and $\omega_c=2.19737$ rad/s. Loop phase is $\tan^{-1}\omega_c-180^\circ$. Phase margin is therefore $\tan^{-1}\omega_c=65.53^\circ$, rounding to 65.5°.
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